Veronika Körber | Searcharxiv

arXiv2026

Triangulations and Maximal Cross-Ratio Degrees

The cross-ratio degree problem is about counting rational curves with $n$ marked points satisfying $n-3$ cross-ratio conditions. This problem has a tropical analogue which provides the same number, as shown by a correspondence theorem. In general, there are no closed formulas for this counting problem. In the special case of cross-ratio conditions given by triangulations, a formula was found by Silversmith via techniques of algebraic geometry. We study the cross-ratio problem given by triangulations in the tropical world. In addition to computing the cross-ratio degree by tropical means, we provide concrete solutions for the counting problem in arbitrary settings, thus answering the question by Silversmith. We also use the tropical recursive algorithm by Goldner to provide a new computational tool to compute cross-ratio degrees. With this, we can find the maximal cross-ratio degrees for $n=9,$ $n=10$ and $n=12,$ where the latter case also makes use of results about Kapranov degrees. Previously, these numbers were only known up to $n=8.$

math.AG↗

arXiv2026

Hamiltonian Cycles in Simplicial and Supersolvable Hyperplane Arrangements

Motivated by the Gray code interpretation of Hamiltonian cycles in Cayley graphs, we investigate the existence of Hamiltonian cycles in tope graphs of hyperplane arrangements, with a focus on simplicial, reflection, and supersolvable arrangements. We confirm Hamiltonicity for all 3-dimensional simplicial arrangements listed in the Grünbaum--Cuntz catalogue. Extending earlier results by Conway, Sloane, and Wilks, we prove that all restrictions of finite reflection arrangements, including all Weyl groupoids and crystallographic arrangements, admit Hamiltonian cycles. Finally, we further establish that all supersolvable hyperplane arrangements and supersolvable oriented matroids have Hamiltonian cycles, offering a constructive proof based on their inductive structure.

math.CO↗